Our work in Chapters 4 and 5 represents the first instance where the grand canonical Brower-Rebbi-Schaich (BRS) algorithm has been applied to the hexagonal Hubbard model (beyond mere proofs of principle), and we have found highly promising results. We emphasize that previously encountered issues related to the computational scaling and ergodicity of the HMC updates have been solved [56]. Our findings are split into two parts, broadly speaking Chapter 4 focuses on electric and Chapter 5 on magnetic properties.
We calculated the single particle gap ∆ as well as all operators that contribute to the anti-ferromagnetic (AFM), anti-ferromagnetic (FM), and charge density wave (CDW) order parameters of the Hubbard Model on a honeycomb lattice. Furthermore we provide a comprehensive analysis of the temporal continuum, thermodynamic and zero-temperature limits for all these quantities.
The favorable scaling of the HMC enabled us to simulate lattices withL >100and to perform a highly systematic treatment of all three limits. The latter limit was taken by means of a finite-size scaling analysis, which determines the critical coupling Uc/κ = 3.835(14) as well as the critical exponentsν = 1.181(43)andβ= 0.898(37).
Depending on which symmetry is broken, the critical exponents of the hexagonal Hubbard model are expected to fall into one of the Gross-Neveu (GN) universality classes [184]. The semimetal-antiferromagnetic Mott insulator (SM-AFMI) transition falls into the GN-Heisenberg SU(2) universality class, as the staggered magnetisationms is described by a vector with three components.
The GN-Heisenberg critical exponents have been studied by means of projection Monte Carlo (PMC) simulations of the hexagonal Hubbard model, by thed= 4−expansion around the upper
critical dimension d, by large N calculations, and by functional renormalization group (FRG) methods. In Table 4.1, we give an up-to-date comparison with our results. Our value forUc/κis in overall agreement with previous Monte Carlo (MC) simulations. For the critical exponents ν andβ, the situation is less clear. Our results forν(assumingz= 1due to Lorentz invariance [184]) agree best with the MC calculation (in the Blankenbecler-Sugar-Scalapino (BSS) formulation) of Ref. [161], followed by the FRG and largeN calculations. On the other hand, our critical exponent ν is systematically larger than most PMC calculations and first-order4−expansion results. The agreement appears to be significantly improved when the4−expansion is taken to higher orders, although the discrepancy between expansions forνand1/νpersists. Finally, our critical exponent βdoes not agree with any results previously derived in the literature. They have been clustering in two regions. The PMC methods and first order 4− expansion yielding values between 0.7 and 0.8, the other methods predicting values larger than 1. Our result lies within this gap at approximately0.9and our uncertainties do not overlap with any of the other results.
Thus, though we are confident to have pinned down the nature of the phase transition and to have performed a thorough analysis of the critical parameters, the values of ν and β remain ambiguous. This is mostly due to a large spread of incompatible results existing in the literature prior to this work. With our values derived by an independent method we add a valuable con-firmation for the critical coupling and some estimations ofν as well as a new but plausible result forβ.
In addition to an unambiguous classification of the character of the quantum critical point (QCP) of the honeycomb Hubbard model, our results demonstrate the ability to perform high-precision calculations of strongly correlated electronic systems using lattice stochastic methods.
A central component of our calculations is the Hasenbusch-accelerated HMC algorithm, as well as other state-of-the-art techniques originally developed for lattice QCD. This has allowed us to push our calculations to system sizes which are, to date, still the largest that have been performed, up to 102×102 unit cells (or 20,808 lattice sites), which reaches a physically realistic size in the field of carbon-based nano-materials. Thus, it may be plausible to put a particular experimental system (a nanotube, a graphene patch, or a topological insulator, for instance) into software, for a direct, first-principles Hubbard model calculation.
There are several future directions in which our present work can be developed that go beyond algorithmic improvements and simulations of yet larger lattices. For instance, while the AFMI phase may not be directly observable in graphene, we note that tentative empirical evidence for such a phase exists in carbon nanotubes [192], along with preliminary theoretical evidence from MC simulations presented in Ref. [55]. The MC calculation of the single-particle Mott gap in a (metallic) carbon nanotube is expected to be much easier, since the lattice dimensionLis determ-ined by the physical nanotube radius used in the experiment (and by the number of unit cells in the longitudinal direction of the tube). As electron-electron interaction (or correlation) effects are expected to be more pronounced in the (1-dimensional) nanotubes, the treatment of flat graphene as the limiting case of an infinite-radius nanotube would be especially interesting. Strong correla-tion effects could be even more pronounced in the (0-dimensional) carbon fullerenes (buckyballs), where we are also faced with a fermion sign problem due to the admixture of pentagons into the otherwise-bipartite honeycomb structure [7]. This particular sign problem has the unusual prop-erty of vanishing as the system size becomes large, as the number of pentagons in a buckyball is
fixed by its Euler characteristic to be exactly12, independent of the number of hexagons.
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